A car is parked 20.0 m directly south of a railroad...

A car is parked $20.0 \mathrm{~m}$ directly south of a railroad crossing. A train is approaching the crossing from the west, headed directly east at a speed of $55.0 \mathrm{~m} / \mathrm{s}$. The train sounds a short blast of its $289-\mathrm{Hz}$ horn when it reaches a point $20.0 \mathrm{~m}$ west of the crossing. What frequency does the car's driver hear when the horn blast reaches the car? The speed of sound in air is $343 \mathrm{~m} / \mathrm{s}$. (Hint: Assume that only the component of the train's velocity that is directed toward the car affects the frequency heard by the driver.)

Key Concepts

Doppler Effect

The Doppler effect is a phenomenon where the observed frequency of a wave changes due to the motion of the source relative to the observer. When the source is moving toward the observer, the wavefronts are compressed, leading to an increase in the observed frequency; when the source is moving away, the wavefronts are stretched out, resulting in a lower frequency. This concept is essential in calculating how motion impacts perceived sound or light frequencies.

Vector Decomposition of Velocity

Vector decomposition involves breaking down a velocity vector into its components along specified directions. In problems where only a part of the velocity affects the outcome—such as the component directed toward the observer—the process of decomposition helps isolate that effective part. By using trigonometric functions, one can determine how much of the velocity contributes along the line of sight between the source and the observer.

Relative Motion and Geometry

Understanding the geometry of the positions and motion of objects is critical when analyzing the relative motion between a source and an observer. This typically involves using coordinate systems to determine distances and directions, and applying trigonometric relationships to determine angles between velocity vectors and the line connecting the source and observer. Such geometric reasoning is key to finding the correct component of motion that affects phenomena like the Doppler shift.

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